Page 76 - Vector Analysis

72 CHAPTER 3. Multiple Integrals

Corollary 3.7. Let A Ď Rn be a bounded subset, and f : A Ñ R be a bounded function. If
P1 and P2 are partitions of A, then

L(f, P1) ď U (f, P2) .

Proof. Let P be the common refinement of P1 and P2. Then Proposition 3.6 implies that

L(f, P1) ď L(f, P) ď U (f, P) ď U (f, P2) . ˝

Corollary 3.8. Let A Ď Rn be a bounded subset, and f : A Ñ R be a bounded function.

Then

żż

f (x)dx ď f (x)dx .

AA

Proof. Noting that for each given partition P of A, L(f, P) is a lower bounded for all
possible upper sum; thus

ż for all partitions P of A

L(f, P) ď f (x)dx

A

żż ˝

which further implies that f (x)dx ď f (x)dx .

AA

Proposition 3.9 (Riemann’s condition). Let A Ď Rn be a bounded set, and f : A Ñ R be
a bounded function. Then f is Riemann integrable over A if and only if

@ ε ą 0, D a partition P of A Q U (f, P) ´ L(f, P) ă ε .

Proof. “ñ” Let ε ą 0 be given. By the definition of infimum and supremum, there exist
partition P1 and P2 of A such that

żε and żε
A f (x) dx ´ 2 ă L(f, P2) A f (x) dx + 2 ą U (f, P1) .

Let P be a common refinement of P1 and P2. Since f is Riemann integrable over A,

żż

f (x)dx = f (x)dx; thus Proposition 3.6 implies that

AA

U (f, P) ´ L(f, P) ď U (f, P1) ´ L(f, P2)
ż ε (ż ε)

ă f (x) dx + ´ f (x) dx ´ = ε .
A 2A 2